Structure of Optical Vortices

Jennifer E. Curtis and David G. Grier
Dept. of Physics, James Franck Institute
and Institute for Biophysical Dynamics
The University of Chicago, Chicago, IL 60637

Date: January 22, 2003

Abstract:

Helical modes of light can be focused into toroidal optical
traps known as optical vortices, which are capable of localizing and applying
torques to small volumes of matter. Measurements of optical vortices
created with the dynamic holographic optical tweezer technique
reveal an unsuspected dependence of their structure and
angular momentum flux on their helicity. These measurements also
provide evidence for a novel optical ratchet potential in practical
optical vortices.

Beams of light with helical wavefronts
focus to rings, rather than points, and
also carry orbital angular momentum (1,3,2)
that they
can transfer to illuminated objects
(6,4,5,3).
When focused strongly enough, such helical modes
form toroidal optical traps
known as optical vortices (8,4,7),
whose properties present novel opportunities for scientific
research
and technological applications.
For example, optical vortices should be ideal actuators for
microelectromechanical systems (MEMS) (9), and
arrays of optical vortices (10)
have shown a promising ability to assemble
colloidal particles into mesoscopic pumps for microfluidic systems.
All such applications will require a comprehensive understanding of the
intensity distribution and angular momentum flux within optical vortices.
This Letter describes measurements of the structure of optical vortices
created with the dynamic holographic optical tweezer
technique (10), and of their ability to exert torques on trapped materials.
These measurements reveal qualitative discrepancies with predicted behavior,
which we explain on the basis of scalar diffraction theory.

Figure 1:
(a) Schematic diagram of dynamic holographic optical tweezers creating
an optical vortex. The SLM imposes the phase
on the incident TEM beam, converting it into a helical beam that is
focused into a optical vortex. The inset phase mask encodes an optical
vortex.
(b) Image of the resulting optical vortex obtained by placing a mirror
in the focal plane.
The central spot is the diffraction-limited focus of an separate coaxial TEM beam
and coincides with the optical axis.
(c) Time-lapse image of a single colloidal sphere
traveling around the optical vortex.

A helical mode
is distinguished by a phase factor
proportional to the polar angle around the beam's axis,

(1)

Here,
is the beam's wavevector,
is the field's radial profile at position ,
and is an integral winding number known as the topological charge.
All phases appear along the beam's axis, , and
the resulting destructive interference cancels the axial intensity.
Similarly, each ray in such a beam has an out-of-phase counterpart with which
it destructively interferes when the beam is brought to a focus.
Constructive interference at a radius from the optical axis
yields a bright ring whose width is comparable to , the
wavelength of light.
The semi-classical approximation further suggests that
each photon in a helical mode carries
orbital angular momentum
(2), so that the beam can exert a torque
proportional to its intensity.

Conventional beams of light can be converted into helical modes
with a variety of mode converters
(11).
Most implementations yield
topological charges in the range
(6,2).
By contrast, dynamic holographic optical tweezers (10)
can create helical modes up to
, and so are ideal for studying how
optical vortices' properties vary with .

Our system, depicted in Fig. 1,
uses a Hamamatsu X7550 parallel-aligned
nematic liquid crystal spatial light modulator (SLM) (12) to imprint
computer-generated patterns of phase shifts
onto the wavefront of a linearly polarized TEM beam at
from a frequency-doubled
Nd:YVO laser (Coherent Verdi).
The modulated wavefront is transferred by a telescope to the back aperture of a
NA oil immersion objective lens mounted in
a Zeiss Axiovert S100TV inverted optical microscope.
The objective lens focuses the
light into optical traps,
in this case a single optical vortex.
The same lens also forms images of trapped particles that are relayed to
an attached video camera through a dichroic
mirror.

The SLM can shift the light's phase to any of 150 distinct levels
in the range
at each 40
wide pixel in a
square array.
Imprinting a discrete approximation to the phase modulation
onto the incident beam,
yields a helical mode with 70 percent efficiency, independent of laser power
over the range studied.
Conversion efficiency is reduced for
by the
SLM's limited spatial resolution.
Fig. 1(b) is a digital image of
an optical vortex reflected
by a mirror placed in the objective's focal plane.
The unmodified portion of the TEM beam travels along the
optical axis and comes to a focus in the center of the field of view.
This conventional beam does not overlap with the optical vortex in
the focal plane and so does not affect our observations.
Fig. 1(c) shows a time-lapse multiple exposure
of a single 800 nm diameter colloidal polystyrene sphere trapped on
the optical vortex's circumference in an 85
thick layer of water between a
coverslip and a microscope slide.
Angular momentum transferred from the optical vortex drives the sphere once around
the circumference in a little under 2 sec at
an applied power of 500 mW.
The image shows 11 stages in its transit at 1/6 sec intervals.
We studied the same particle's motions at different topological charges and applied
powers to establish how helicity influences optical vortices'
intensity distribution and local angular momentum flux.

Observing that a single particle
translates around the circumference of a
linearly polarized optical
vortex distinguishes the angular momentum carried by a helical
beam of light from that carried by circularly
polarized light.
The latter causes an absorbing particle to spin on its own axis.
Observing that the particle instead translates around the optical axis
demonstrates that the angular momentum density in a helical beam
results from a transverse component of the linear momentum density, as
predicted (6,2).

Figure 2:
Dependence of optical vortex's radius on topological charge .
The dashed line is predicted by Eq. (5) with no free parameters.
Inset: Azimuthally averaged intensity at from the image in
Fig. 1(b).

Most observed characteristics of optical vortices have been
interpreted in terms of the properties of Laguerre-Gaussian (LG)
eigenmodes of the paraxial Helmholtz equation (6,5).
These have a radial dependence

(2)

where
is a generalized Laguerre polynomial with radial
index , and
is the beam's radius (1).
An LG mode with appears as a ring of light
whose radius depends on topological charge as
(13).
Practical optical vortices, including the example in Fig. 1(b),
also appear as rings of light
and so might be expected to scale in the same way (6).
However, the data in Fig. 2 reveal qualitatively
different behavior.
We obtain from digitized images
such as Fig. 1(b) by averaging over angles
and locating the radius of peak intensity.
Projecting different values of reveals
that scales linearly with the topological charge, and not as
.

This substantial discrepancy can be explained by considering
how the phase-modulated beam propagates through the optical train.
The field in the focal plane of a lens of focal length
is related in scalar diffraction theory
to the field at the input aperture
(and thus at the face of the SLM) through a Fourier transform (14).
Transforming the helical beam first over angles yields

(3)

where is the -th order
Bessel function of the first kind,
and is the radius of the optical train's effective aperture.
For our system,
.
Setting
for a uniform illumination yields

Even if the mode converter created a pure LG mode, the optical trapping system's
limited aperture, , still would yield a superposition of radial eigenmodes at the
focal plane (2), and a comparable linear dependence of on .
The superposition of higher- modes in our system is evident in the hierarchy of
diffraction fringes surrounding the principal maximum in Fig. 1(b).

Figure 3:
Time required for colloidal sphere
to complete one circuit of an optical vortex.
Dashed curves indicate scaling predicted by Eq. (6);
Solid curves result from fits to Eqs. (12) and (13).
(a) Dependence of
on topological
charge for
. Inset: corrugated intensity distribution
around one quarter of the circumference of
an optical vortex measured at reduced
intensity, compared with calculated pattern at at the same scale.
(b) Dependence of on applied power for .
The dotted curve
includes the influence of a localized hot spot through Eq. (14).
Inset: Potential energy landscape calculated from fits to data in (a) and (b)
for and
.

This linear dependence on leads to scaling predictions
for optical vortices' optomechanical properties with which we can
probe the nature of the angular momentum carried by helical modes.
In particular, a wavelength-scale
particle trapped on the circumference of an optical vortex
is illuminated with an intensity
,
where is the power of the input beam, assumingthat the photon flux is
spread uniformly around the vortex's circumference in a band roughly
thick (see Fig. 2).
If we assume that each scattered photon transfers an angular
momentum proportional to
, then
the particle's tangential speed should be proportional
to
,
and the time required to make one circuit of the optical vortex
should scale as

(6)

If the radius had scaled as
, then the
particle's speed would have been independent of , and the period
would have scaled as
.
Instead, for
, we expect
.

The data in
Fig. 3(a) show that
does indeed scale according to Eqs. (5) and (6) for
larger values of .
For , however, the period is systematically larger than predicted.
Similarly, scales with as predicted for lower
powers, but increases as increases.
In other words, the particle moves slower the harder it is pushed.
This unexpected effect can be ascribed to the detailed
structure of optical vortices created with pixellated diffractive
optical elements;
the mechanism presents new opportunities for studying Brownian
transport in modulated potentials.

When projected onto our objective lens' input pupil, each
of our SLM's effective phase pixels
spans roughly
.
Numerically transforming such an apodized
beam reveals a pattern of intensity corrugations,
as shown in Fig. 3(a).
These establish
a nearly sinusoidal potential through which the
particle is driven by the local angular momentum flux.
We model the intensity's dependence on arclength around the ring as

(7)

where is the depth of the modulation, and
is its wavenumber.
For
,
is approximately independent of .

This modulated intensity exerts two tangential forces on the trapped
sphere.
One is due to the transferred angular momentum,

(8)

where we assume a local angular momentum flux of
per photon.
The prefactor includes such geometric factors
as the particle's scattering cross-section.
The other is an optical gradient force due to the polarizable particle's
response to local intensity gradients:

(9)

where sets the relative strength of the gradient force.
Combining Eqs. (8) and (9) yields the tangential force

(10)

where we have omitted an irrelevant phase angle, and
where
.
Even if is much smaller than
unity, both and can be much larger.
In that case, reducing at fixed power increases the depth of the
modulation relative to the thermal energy scale ,
and the particle can become hung up in the local potential minima.
The modulated potential thus increases the effective drag.

More formally, a particle's motion along an inclined sinusoidal potential with strong
viscous damping is described by the Langevin equation

(11)

where is the viscous drag coefficient and is a
zero-mean random thermal force.
The associated mobility
may be expressed as (15)

(12)

where
is the topological charge at which the modulation reaches at power .
Given this result, the transit time for one cycle should be

(13)

where
is the expected period for
at in the absence of modulation.
The solid curve in Fig. 3(a) is a fit to
Eqs. (12) and (13) for , and .
The results,
,
,
and , are consistent with the strongly modulated
potential shown in the inset to Fig. 3(b).
Rather than smoothly processing around the optical vortex, the particle
instead makes thermally activated hops between potential wells in a
direction biased by the optical vortex's torque.

Replacing
with in Eq. (12)
yields an analogous result for the period's dependence on applied
power for fixed , as shown in Fig. 3(b).
Here,
is the power at
which the modulation reaches .
Using and
obtained from Fig. 3(a), we find that the sphere's motions
above
are slower even than our model predicts.
The period's divergence at high power is due to a
localized ``hot spot'' on the optical vortex
resulting from aberrations in our optical train.
Such hot spots have confounded previous attempts to study single-particle
dynamics in helical beams (3).
Because hot spots also deepen with increasing power,
they retain particles with exponentially increasing residence times.
The total transit time becomes
(15)

(14)

The data in Fig. 3(b)
are consistent with
and
.
Localization in hot spots becomes comparable to corrugation-induced drag only for
powers above
and so does not affect the data in Fig. 3(a).
Consequently, these data offer insights into the nature of the helical beam's
angular momentum density.

In particular, the simple scaling relation, Eq. (6) is remarkably
successful at describing a single particle's motions around an optical vortex
over a wide range of topological charges.
This success
strongly supports the contention that
each photon contributes
to the
local angular momentum flux of a helical beam of light, and not only to
the beam's overall angular momentum density.
It hinges on our observation that the radius of
a practical optical vortex scales linearly with its
topological charge.
The corrugations in apodized optical vortices broadens this system's interest.
Not only do they
provide a realization of the impossible staircase in M. C. Escher's
lithograph ``Ascending and Descending'',
but they also offer a unique opportunity to study
overdamped transport on tilted sinusoidal potentials.
As a practical Brownian ratchet, this system promises insights germane to such related
phenomena as transport by molecular motors,
voltage noise in Josephson junction arrays,
and flux flow in type-II superconductors.
Preliminary observations of multiple particles on an optical vortex
also suggest opportunities to study transitions
from jamming to cooperativity with increasing occupation.

This work was supported by a grant from Arryx, Inc., and in part by
the MRSEC program of the National Science Foundation through Grant DMR-9880595
Equipment was purchased with funds from the W. M. Keck Foundation, and the
spatial light modulator was made available by Hamamatsu Corp.